Correction of CT images for truncated or incomplete projections

ABSTRACT

A data consistency condition is derived for an array of attenuation values acquired with a fan-beam x-ray CT system. Using this data consistency condition, estimates of selected attenuation values can be calculated from the other attenuation values acquired during the scan. Such estimates reduce artifacts caused truncated data and by loss of data due to x-ray absorption.

This invention was made with government support under Grant No. 1R21EB001683-01 awarded by the National Institute of Health. The UnitedStates Government has certain rights in this invention.

BACKGROUND OF THE INVENTION

The field of the present invention is computed tomography and,particularly, computer tomography (CT) scanners used to produce medicalimages from x-ray attenuation measurements.

As shown in FIG. 1, a CT scanner used to produce images of the humananatomy includes a patient table 10 which can be positioned within theaperture 11 of a gantry 12. A source of highly columinated x-rays 13 ismounted within the gantry 12 to one side of its aperture 11, and one ormore detectors 14 are mounted to the other side of the aperture. Thex-ray source 13 and detectors 14 are revolved about the aperture 11during a scan of the patient to obtain x-ray attenuation measurementsfrom many different angles through a range of at least 180° ofrevolution.

A complete scan of the patient is comprised of a set of x-rayattenuation measurements which are made at discrete angular orientationsof the x-ray source 13 and detector 14. Each such set of measurements isreferred to in the art as a “view” and the results of each such set ofmeasurements is a transmission profile. As shown in FIG. 2A, the set ofmeasurements in each view may be obtained by simultaneously translatingthe x-ray source 13 and detector 14 across the acquisition field ofview, as indicated by arrows 15. As the devices 13 and 14 aretranslated, a series of x-ray attenuation measurements are made throughthe patient and the resulting set of data provides a transmissionprofile at one angular orientation. The angular orientation of thedevices 13 and 14 is then changed (for example, 1°) and another view isacquired. An alternative structure for acquiring each transmissionprofile is shown in FIG. 2B. In this construction, the x-ray source 13produces a fan-shaped beam which passes through the patient and impingeson an array of detectors 14. Each detector 14 in this array produces aseparate attenuation signal and the signals from all the detectors 14are separately acquired to produce the transmission profile for theindicated angular orientation. As in the first structure, the x-raysource 13 and detector array 14 are then revolved to a different angularorientation and the next transmission profile is acquired.

As the data is acquired for each transmission profile, the signals arefiltered, corrected and digitized for storage in a computer memory.These steps are referred to in the art collectively as “preprocessing”and they are performed in real time as the data is being acquired. Theacquired transmission profiles are then used to reconstruct an imagewhich indicates the x-ray attenuation coefficient of each voxel in thereconstruction field of view. These attenuation coefficients areconverted to integers called “CT numbers”, which are used to control thebrightness of a corresponding pixel on a CRT display. An image whichreveals the anatomical structures in a slice taken through the patientis thus produced.

The reconstruction of an image from the stored transmission profilesrequires considerable computation and cannot be accomplished in realtime. The prevailing method for reconstructing images is referred to inthe art as the filtered back projection technique.

Referring to FIG. 3, the proper reconstruction of an image requires thatthe x-ray attenuation values in each view pass through all of theobjects located in the aperture 11. If the object is larger than theacquired field of view, it will attenuate the values in sometransmission profiles as shown by the vertically oriented view in FIG.3, which encompasses the supporting table 10, and it will not attenuatethe values in other transmission profiles as shown by the horizontallyoriented view in FIG. 3. As a result, when all of the transmissionprofiles are back projected to determine the CT number of each voxel inthe reconstructed field of view, the CT numbers will not be accurate.This inaccuracy caused by truncated projection data can be seen in thedisplayed image as background shading which can increase the brightnessor darkness sufficiently to obscure anatomical details.

A similar problem is presented when transmission profiles are affectedby metal objects such as dental filings in the patient being scanned. Inthis situation x-rays passing through the metal object are stronglyabsorbed and the attenuation measurement is very noisy causing strongartifacts in the reconstructed image.

The data truncation problem and the x-ray absorption problem eachcorrupt the acquired attenuation data set in a unique way. Referring toFIG. 4, as views of the attenuation data are acquired the attenuationvalues 32 in each view are stored on one row of a two dimensional dataarray 33. As indicated by the dashed line 34, each such row ofattenuation data provides a transmission profile of the object to beimaged when viewed from a single view angle. One dimension of the dataarray 33 is determined by the number of views which are acquired duringthe scan and the other dimension is determined by the number of detectorcell signals acquired in each view.

Referring particularly to FIG. 5A, the truncated data problem can bevisualized as a set of contiguous views 36 in the acquired data array 33that are corrupted because they include attenuation information fromobjects (e.g., supporting table, patient's shoulder or arms) outside thefield of view of all the remaining acquired views. On the other hand, asshown in FIG. 5B the absorbed x-ray problem can be visualized as thecorruption of one or more attenuation values in all, or nearly all theacquired views as indicated at 38. In the first problem a select few ofthe acquired views are significantly affected and in the second problemall or nearly all the acquired views are affected in a more limitedmanner.

SUMMARY OF THE INVENTION

The present invention is a method for correcting individual attenuationvalues in fan-beam projections that have been corrupted. Moreparticularly, the present invention is a method which employs a novelfan-beam data consistency condition to estimate individual attenuationmeasurements in one acquired fan-beam projection view from attenuationmeasurements made at the other view angles. Corrupted data acquiredduring a scan is replaced with estimated values produced according tothis method.

A general object of the invention is to replace corrupted x-rayattenuation data acquired during a scan with attenuation data calculatedfrom other, uncorrupted attenuation data acquired during the scan. Theestimated and replaced attenuation data may be one or more entireattenuation profiles as occurs when correcting data truncation problems,or it may be selected attenuation values in one or more attenuationprofiles as occurs when correcting x-ray absorption problems.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of an x-ray CT system which employs thepresent invention;

FIGS. 2A and 2B are pictorial representations of a parallel beam andfan-beam scan respectively that may be performed with a CT system;

FIG. 3 is a pictorial representation of a fan-beam acquisition situationwhich results in a data truncation problem that is solved using thepresent invention;

FIG. 4 is a pictorial representation of an attenuation profile acquiredby the system of FIG. 1 and its storage in a data array;

FIGS. 5A and 5B are pictorial representations of the data array of FIG.4 illustrating data that may be corrupted by two problems encounteredwhen scanning subjects with the system of FIG. 1;

FIG. 6 is a graphic representation of an x-ray beam scan which shows thegeometric parameters used to derive data consistency condition;

FIG. 7 is a graphic representation used in the derivation of the dataconsistency condition;

FIG. 8 is a graphic representation showing the relationship of vectors;

FIG. 9 is a block diagram of a preferred embodiment of an x-ray CTsystem which employs the present invention; and

FIG. 10 is a pictorial representation of the data structures that areproduced when practicing the steps of the present invention.

GENERAL DESCRIPTION OF THE INVENTION

It is well known that if all the projection data are summed in one viewof non-truncated parallel-beam projections, the result is a view angleindependent constant. Mathematically, this is a special case (zero-ordermoment) of the so-called Helgason-Ludwig consistency condition ontwo-dimensional Radon transforms. Physically, this condition states thatthe integral of the attenuation coefficients over the whole transmissionprofile should be a view angle independent constant. This dataconsistency condition (DCC) plays an important role in correcting theX-ray CT image artifacts when some projection data are missing inparallel beam scans. In practice, this may happen when a portion of ascanned object is positioned outside the scan field-of-view (FOV)defined by a CT scanner.

A novel data consistency condition is derived here which enablesestimation of attenuation values for fan-beam projections. It will becalled a fan-beam data consistency condition (FDCC). The new FDCCexplicitly gives an estimation of the projection data for a specific rayby filtering all the available fan-beam projections twice. To derive theFDCC, the following definition of a fan-beam projection g[{right arrowover (r)}, {right arrow over (y)}(t)] is used as the starting point

$\begin{matrix}{{g\left\lbrack {\overset{\rightarrow}{r},{\overset{\rightarrow}{y}(t)}} \right\rbrack} = {\int_{0}^{\infty}\ {{\mathbb{d}{{sf}\left\lbrack {{\overset{\rightarrow}{y}(t)} + {s\overset{\rightarrow}{r}}} \right\rbrack}}.}}} & (1)\end{matrix}$The source trajectory vector {right arrow over (y)}(t) is parameterizedby a parameter t, and r is a vector starting from the source position tothe image object as shown in FIG. 6. The vector {right arrow over(y)}(t) denotes a source position and the vector {right arrow over (r)}represents a vector from the x-ray source to the imaged object. In alaboratory coordinate system o—xy, the vector {circumflex over (γ)}(t)is parameterized by a polar angle t, and the vector {circumflex over(r)} is parameterized by polar angle φ. The fan angle γ is also definedfrom the iso-ray. All the angles have been defined according to acounterclockwise convention. The image function ƒ({right arrow over(x)}) is assumed to have a compact support Ω, i.e., it is non-zero onlyin a finite spatial region. Throughout this discussion, a vector will bedecomposed into its magnitude and a unit vector, e.g. {right arrow over(r)}=r {circumflex over (r)}. Although a circular scanning geometry isshown in FIG. 6, the present invention may be employed in any geometryand a general vector notation is used herein to reflect this fact.

Eq. (1) defines a homogeneous extension of the conventional fan-beamprojections {overscore (g)}[{circumflex over (r)}, {right arrow over(y)}(t)]. That is

$\begin{matrix}{{g\left\lbrack {\overset{\rightarrow}{r},{\overset{\rightarrow}{y}(t)}} \right\rbrack} = {{\frac{1}{r}{\overset{\_}{g}\left\lbrack {\hat{r},{\overset{\rightarrow}{y}(t)}} \right\rbrack}} = {\frac{1}{r}{\int_{0}^{\infty}\ {{\mathbb{d}{{sf}\left\lbrack {{\overset{\rightarrow}{y}(t)} + {s\hat{r}}} \right\rbrack}}.}}}}} & (2)\end{matrix}$A Fourier transform G[{right arrow over (k)}, {right arrow over (y)}(t)]of the generalized fan-beam projection g[{right arrow over (r)},{rightarrow over (y)}(t)] with respect to variable {right arrow over (r)} isdefined as

G ⁡ [ k → , y → ⁡ ( t ) ] = ∫ 2 ⁢ ⁢ ⅆ 2 ⁢ rg ⁡ [ r → , y → ⁡ ( t ) ] ⁢ exp ⁡ ( -2 ⁢ π ⁢ ⁢ i ⁢ k → · r → ) . ( 3 )Note that this Fourier transform is local, since the Fourier transformis taken with respect to the vectors that emanate from a source positionlabeled by {right arrow over (y)}(t).

By choosing a separate polar coordinate system for vectors {right arrowover (k)} and {right arrow over (r)} and using Eq. (2), the Fouriertransform G[{right arrow over (k)}, {right arrow over (y)}(t)] can befactorized into the product of a divergent radial component

$\frac{1}{k}$and an angular component {overscore (G)}[{circumflex over (k)},{overscore (y)}(t)]. That is

$\begin{matrix}{{G\left\lbrack {\overset{\rightarrow}{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack} = {\frac{1}{k}{{\overset{\_}{G}\left\lbrack {\hat{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack}.}}} & (4)\end{matrix}$Here {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)]is similarly defined by Eq. (3), but the vector {right arrow over (k)}is replaced by a unit vector {circumflex over (k)}.

A connection between G[{right arrow over (k)}, {overscore (y)}(t)] andthe Fourier transform {tilde over (ƒ)}({overscore (k)}) of the objectfunction ƒ({right arrow over (x)}) can be established by inserting Eq.(1) and Eq. (2) into Eq. (3). The result is

$\begin{matrix}{{G\left\lbrack {\overset{\rightarrow}{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack} = {\frac{1}{k}{\int_{0}^{\infty}\ {{\mathbb{d}s}{\overset{\sim}{f}\left( {s\hat{k}} \right)}{{\exp\left\lbrack {{\mathbb{i}2\pi}\; s{\hat{k} \cdot {\overset{\rightarrow}{y}(t)}}} \right\rbrack}.}}}}} & (5)\end{matrix}$Compared to Eq. (4), the angular component {overscore (G)}[{circumflexover (k)}{circumflex over (,)} {right arrow over (y)}(t)] is given by

$\begin{matrix}{{\overset{\_}{G}\left\lbrack {\hat{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack} = {\int_{0}^{\infty}\ {{\mathbb{d}k}{\overset{\sim}{f}\left( {k\hat{k}} \right)}{\exp\left\lbrack {{\mathbb{i}}\; 2\pi\; k{\hat{k} \cdot {\overset{\rightarrow}{y}(t)}}} \right\rbrack}}}} & (6)\end{matrix}$The presence of the integral in Eq. (6) indicates that the function{overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] doesnot explicitly depend on the vector {right arrow over (y)}(t). Rather,the function {overscore (G)}[{circumflex over (k)}, {right arrow over(y)}(t)] depends directly on the projection of vector {right arrow over(y)}(t) onto a given unit vector {circumflex over (k)}. Therefore, it isappropriate to introduce a new variablep={circumflex over (k)}·{right arrow over (y)}  (7)and rebin the data {overscore (G)}[{circumflex over (k)}, {right arrowover (y)}(t)] into G_(r)({circumflex over (k)}, p) by the followingrelation{overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)]=G_(r)({circumflex over (k)}, p={circumflex over (k)}·{right arrow over(y)}( t)).  (8)Using Eqs. (7) and (8), Eq. (6) can be rewritten as

$\begin{matrix}{{G_{r}\left( {\hat{k},p} \right)} = {\int_{0}^{\infty}\ {{\mathbb{d}k}{\overset{\sim}{f}\left( {k\hat{k}} \right)}{\exp\left( {{\mathbb{i}}\; 2\;\pi\;{kp}} \right)}}}} & (9)\end{matrix}$

From the definition given in Eq. (3), it is apparent that the functionGr (k, p) is a complex function, and thus has both an imaginary part anda real part. In general, the imaginary part and the real part of thefunction G_(r)({circumflex over (k)}, p) are not correlated with oneanother. However, Eq. (9) imposes a strong constraint relating theimaginary and real parts of the function G_(r)({right arrow over (k)},p). To better understand this hidden constraint, the following fact isinsightful. The variable {right arrow over (k)} is the magnitude of thevector {circumflex over (k)}, and thus it is intrinsically anon-negative number. Using this property, Eq. (9) may be extended to therange of (−∞, +∞) by introducing a following function

$\begin{matrix}{{Q\left( {\hat{k},k} \right)} = \left\{ \begin{matrix}{\overset{\sim}{f}\left( {k\hat{k}} \right)} & {k \geq 0} \\0 & {k < 0}\end{matrix} \right.} & (10)\end{matrix}$In other words, function Q({circumflex over (k)}, k) is a causalfunction with respect to the variable k. Using the functionQ({circumflex over (k)}, k), Eq. (9) can be written as:

$\begin{matrix}{{G_{r}\left( {\hat{k},p} \right)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{{kQ}\left( {\hat{k},k} \right)}}{\exp\left( {{\mathbb{i}2\pi}{pk}} \right)}}}} & (11)\end{matrix}$Therefore, a standard inverse Fourier transforms yields:

$\begin{matrix}{{Q\left( {\hat{k},k} \right)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{{pG}_{r}\left( {\hat{k},p} \right)}}{\exp\left( {{- {\mathbb{i}2\pi}}\;{pk}} \right)}}}} & (12)\end{matrix}$

Note that function Q({circumflex over (k)},k) satisfies the causalstructure dictated by Eq. (10). This fact implies that, for negative k,the integral in Eq. (12) should universally converge and that the valueof the integral should be zero. Therefore, the integral must be done inthe upper half of the complex p—plane. In addition, according to a knownmathematical theorem, the causal structure in Eq. (10) and (12) requiresthe function Gr (k, p) to be analytical in the upper half of the complexp—plane. An intuitive argument is also beneficial in order todemonstrate that the function G_(r)({circumflex over (k)}, p) isanalytical in the upper half of the complex p—plane. For negative k, thecontour of integration for Eq. (12) should be closed by a largesemicircle that encloses the upper half of the complex plane as shown inFIG. 7. By Cauchy's theorem, the integral will vanish ifG_(r)({circumflex over (k)}, p) is analytic everywhere in the upper halfplane. Thus, the intuitive argument also leads to the conclusion thatthe function G_(r)({circumflex over (k)}, p) is analytical in the upperhalf of the complex p—plane.

The complex function G_(r)({circumflex over (k)}, p) may be separatedinto a real part and an imaginary part asG _(r)({circumflex over (k)}, p)=Re G _(r)({circumflex over (k)},p)+i ImG _(r)({circumflex over (k)}, p).  (13)The causal structure implied in Eq. (10) and the concomitant analyticalstructure shown in FIG. 7 require that the real part and imaginary partof the function G_(r)({circumflex over (k)}, p) are mutually linked inthe following way:

$\begin{matrix}{{{{Re}G}_{r}\left( {\hat{k},p} \right)} = {\frac{1}{\pi}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}p^{\prime}}\frac{{Im}\;{G_{r}\left( {\hat{k},p^{\prime}} \right)}}{p^{\prime} - p}}}}} & \left( \text{14a} \right)\end{matrix}$

$\begin{matrix}{{{{{Im}G}_{r}\left( {\hat{k},p} \right)} = {\frac{1}{\pi}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}p^{\prime}}\frac{{Re}\;{G_{r}\left( {\hat{k},p^{\prime}} \right)}}{p^{\prime} - p}}}}},} & \left( \text{14b} \right)\end{matrix}$where the symbol

represents Cauchy principal value. In other words, the imaginary partand real part of the function G_(r)({circumflex over (k)},p) are relatedto each other by a Hilbert transform.

The imaginary part and real part of function {overscore (G)}[{circumflexover (k)}, {right arrow over (y)}(t)] have been explicitly calculated.The final results are

$\begin{matrix}{{{Re}{\overset{\_}{G}\left\lbrack {\hat{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack}} = {\frac{1}{2}{\int_{0}^{2\pi}\ {{\mathbb{d}{{\varphi\delta}\left( {\hat{k} \cdot \hat{r}} \right)}}{\overset{\_}{g}\left\lbrack {\hat{r},{\overset{\rightarrow}{y}(t)}} \right\rbrack}}}}} & (15)\end{matrix}$

$\begin{matrix}{{{Im}{\overset{\_}{G}\left\lbrack {\hat{k},{\overset{\rightarrow}{y}(t)}} \right\rbrack}} = {{- \frac{1}{2\pi}}{\int_{0}^{2\pi}\ {{\mathbb{d}\varphi}{\frac{\overset{\_}{g}\left\lbrack {\hat{r},{\overset{\_}{y}(t)}} \right\rbrack}{\hat{k} \cdot \hat{r}}.}}}}} & (16)\end{matrix}$

Here the angular variable φ is the azimuthal angle of the unit vector{circumflex over (r)}, i.e. {circumflex over (r)}=(cosφ, sinφ). Animportant observation is that there is a Dirac δ—function in Eq. (15).As shown in FIG. 8, for a given unit vector {circumflex over (k)} andsource position {right arrow over (y)}(t), the real part Re{overscore(G)}[{circumflex over (k)}, {right arrow over (y)}(t)] is completelydetermined by a single ray along the direction {circumflex over(r)}={right arrow over (k)}^(⊥). Note that the clockwise convention hasbeen chosen to define the unit vector {circumflex over (r)} from a givenunit vector {circumflex over (k)}. Thus, the real part is given by:Re{overscore (G)}[{circumflex over (k)}, {right arrow over(y)}(t)]=½{overscore (g)}[{circumflex over (r)}={overscore (k)} ^(⊥) ,{right arrow over (y)}(t)]=Re G _(r)({circumflex over (k)},p={circumflex over (k)}·{overscore (y)}(t))  (17)This equation can also be written as{overscore (g)}[{circumflex over (r)}, {right arrow over (y)}(t)]=2 Re G_(r)({circumflex over (k)}={circumflex over (r)} ^(⊥) , p={circumflexover (r)} ^(⊥) ·{right arrow over (y)}(t))  (18)Using Eq. (14) and Eq. (16), the following consistency condition on thefan-beam projection data may be derived

$\begin{matrix}{{\overset{\_}{g}\left\lbrack {{\hat{r}}_{0},{\overset{\rightarrow}{y}\left( t_{0} \right)}} \right\rbrack} = {\frac{2}{\pi}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}p^{\prime}}\frac{{Im}\;{G_{r}\left( {{\hat{r}}_{0}^{\bot},p^{\prime}} \right)}}{p^{\prime} - {{\hat{r}}_{0}^{\bot} \cdot {\overset{\rightarrow}{y}\left( t_{0} \right)}}}}}}} & (19)\end{matrix}$

$\begin{matrix}{{{{Im}G}_{r}\left( {{\hat{r}}_{0}^{\bot},{p^{\prime} = {{\hat{r}}_{0}^{\bot} \cdot {\overset{\rightarrow}{y}(t)}}}} \right)} = {{- \frac{1}{2\pi}}{\int_{0}^{2\pi}\ {{\mathbb{d}\varphi}{\frac{\overset{\_}{g}\left\lbrack {\hat{r},{\overset{\rightarrow}{y}(t)}} \right\rbrack}{{\hat{r}}_{0}^{\bot} \cdot \hat{r}}.}}}}} & (20)\end{matrix}$

In order to obtain one specific attenuation profile of projection data{overscore (g)}[{circumflex over (r)}₀, {overscore (y)}(t₀)] from Eq.(19), all the possible values of Im G_(r)({circumflex over (k)}, p) arerequired at the specific orientation {circumflex over (k)}={circumflexover (r)}₀ ^(⊥). Therefore, it is important to have a scanning path thatfulfills at least the short-scan requirement, viz. angular coverage ofthe source trajectory of 180°+fan angle. Thus, an individual projectionat a specific view angle is linked to the projection data measured fromall the different view angles via Eqs. (19) and (20). In other words, anindividual attenuation profile can be estimated from all the availableprojection data. The novel fan-beam data consistency condition (FDCC) inEq. (19) and Eq. (20) is the basis for the data correction methodaccording to the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

With initial reference to FIG. 9, a computed tomography (CT) imagingsystem 110 includes a gantry 112 representative of a “third generation”CT scanner. Gantry 112 has an x-ray source 113 that projects a fan-beamof x-rays 114 toward a detector array 116 on the opposite side of thegantry. The detector array 116 is formed by a number of detectorelements 118 which together sense the projected x-rays that pass througha medical patient 115. Each detector element 118 produces an electricalsignal that represents the intensity of an impinging x-ray beam andhence the attenuation of the beam as it passes through the patient.During a scan to acquire x-ray projection data, the gantry 112 and thecomponents mounted thereon rotate about a center of rotation 119 locatedwithin the patient 115.

The rotation of the gantry and the operation of the x-ray source 113 aregoverned by a control mechanism 120 of the CT system. The controlmechanism 120 includes an x-ray controller 122 that provides power andtiming signals to the x-ray source 113 and a gantry motor controller 123that controls the rotational speed and position of the gantry 112. Adata acquisition system (DAS) 124 in the control mechanism 120 samplesanalog data from detector elements 18 and converts the data to digitalsignals for subsequent processing. An image reconstructor 125, receivessampled and digitized x-ray data from the DAS 124 and performs highspeed image reconstruction according to the method of the presentinvention. The reconstructed image is applied as an input to a computer126 which stores the image in a mass storage device 129.

The computer 126 also receives commands and scanning parameters from anoperator via console 130 that has a keyboard. An associated cathode raytube display 132 allows the operator to observe the reconstructed imageand other data from the computer 126. The operator supplied commands andparameters are used by the computer 126 to provide control signals andinformation to the DAS 124, the x-ray controller 122 and the gantrymotor controller 123. In addition, computer 126 operates a table motorcontroller 134 which controls a motorized table 136 to position thepatient 115 in the gantry 112.

The fan-beam data consistency condition (FDCC) derived generally aboveis applied to this preferred geometry by restricting the motion of thex-ray source {right arrow over (y)}(t) to a circle centered at theorigin “0” with a radius R. The scanning path is parameterized by apolar angle t shown in FIG. 6. Therefore, we have the followingparameterization of the source trajectory{right arrow over (y)}(t)=R(cos t, sin t).  (21)In addition, it is also useful to consider the followingparameterizations for the unit vectors r, {circumflex over (r)}₀, and{circumflex over (r)}₀ ^(⊥) in the laboratory coordinate system:{circumflex over (r)}=(cos φ, sin φ),  (22){circumflex over (r)} ₀=(cos φ₀, sin φ₀),  (23){circumflex over (r)}₀ ^(⊥)=(−sin φ₀, cos φ₀)  (24)For convenience, the notation g_(m) (γ, t) is used to describe themeasured fan-beam projections with an equi-angular curved detector. Theprojection angle γ is in the range

$\left\lbrack {{- \frac{\gamma_{m}}{2}},\frac{\gamma_{m}}{2}} \right\rbrack$where γ_(m) is the fan angle. By definition,g _(m)(γ,t)={overscore (g)}[{circumflex over (r)}, {right arrow over(y)}(t)]  (25)with the following relation between φ and γφ±π+t+γ  (65)

In practice, it is beneficial to introduce the following definitions:Im G _(r)({circumflex over (r)} ₀ ^(⊥) , p′)= F _(p)(φ₀ , p′)= F _(t)(φ₀, t)  (27)p′=r ₀ ^(⊥) ·{right arrow over (y)}(t)=R sin(t−φ ₀)  (28)φ₀ =π+t ₀+γ₀  (29)In the second equality in Eq. (27), a data rebinning has been introducedvia Eq. (28).

Using these definitions, the FDCC of Eq. (19) and Eq. (20) for thisgeometry may be expressed as follows:

$\begin{matrix}{{g_{m}\left( {\gamma_{0},t_{0}} \right)} = {\frac{2}{\pi}{\int_{- \infty}^{+ \infty}{{\mathbb{d}p^{\prime}}\frac{1}{p_{0} - p^{\prime}}{F_{p}\left( {\varphi_{0},p^{\prime}} \right)}}}}} & (30)\end{matrix}$and

$\begin{matrix}{{{F_{p}\left( {\varphi_{0},p^{\prime}} \right)} = {{F_{t}\left( {\varphi_{0},t} \right)} = {\frac{1}{2\pi}{\int_{{- \gamma_{m}}/2}^{{+ \gamma_{m}}/2}\ {{\mathbb{d}\gamma}\frac{1}{\sin\left( {\varphi_{0} - t - \gamma} \right)}{g_{m}\left( {\gamma,t} \right)}}}}}},} & (31)\end{matrix}$where the number p₀ in Eq. (30) is given by:p ₀ =R sin γ_(o).  (32)

Eqs. (30) and (31) explicitly relate the attenuation data in a singleprojection to the measured attenuation data in all of the other viewangles. Given a desired projection attenuation value labeled byparameters γ₀ and t₀, the numerical procedure to estimate this specificattenuation value may be summarized in the following three steps:

-   Step 1: For each of the other view angles, filter the measured data    by a filtering kernel

$\frac{1}{\sin\;\gamma}$to obtain F₁(φ₀,t) as set forth in Eq. (31).

-   Step 2: Rebin the filtered data F_(l)(φ₀,t) into F_(p)(φ_(o), p′) as    set forth in Eq. (28).-   Step 3: Filter the rebinned data F_(p)(φ₀, p′) by a Hilbert kernel

$\frac{1}{p^{\prime}}$to obtain the estimated projection data g_(m)(γ₀, t₀) as set forth inEq. (30).

This process is implemented by a program executed by the computer 126after the scan is completed and the acquired attenuation data g_(m)(γ₀,t₀) is stored in data array 33. As shown in FIG. 10, the above step 1 isperformed on the entire data set 140 to produce data set F_(t)(φ₀, t)which is stored as array 142. This data set is rebinned as describedabove in step 2 to form F_(p)(φ₀, p′) which is stored in data array 144.The attenuation values g_(m)(γ₀, t₀) at any view angle t₀ can then beestimated using the data in array 144 and Eq. (30) as described above instep 3. It can be appreciated that any acquired attenuation profile canbe estimated in this manner in its entirety, or only a particularattenuation value therein may be estimated. Thus, in the truncated dataproblem illustrated in FIG. 5A, the views 36 in the acquired data array33 are replaced with estimated values, whereas in the absorbed x-rayproblem illustrated in FIG. 5B, the corrupted attenuation values 38 inthe data array 33 are replaced with estimated values.

It should be apparent that the method can be repeated using thecorrected attenuation data array 33 to further improve the results. Suchan iterative process is normally not necessary when only a small amountof the acquired data is corrupted, but further iterations are requiredas the proportion of corrupted data increases.

While the present invention is described with reference to fan-beamx-ray CT systems, it is also applicable to other imaging modalities suchas radiation therapy systems and PET/CT systems. Projection dataacquired with a fan, or divergent, beam may be estimated using thepresent invention where projection data is acquired from the samesubject at a sufficient number of other projection angles.

1. A method for correcting selected attenuation values in an array of attenuation values acquired with a fan-beam x-ray CT system, the steps comprising: a) filtering the attenuation values in the array with a first filtering kernel; b) rebinning the filtered attenuation values; and c) calculating estimates of the selected attenuation values by filtering the rebinned attenuation values with a second filtering kernel.
 2. The method as recited in claim 1 in which the second filtering kernel is a Hilbert kernel.
 3. The method as recited in claim 1 in which the first filtering kernel is 1/sin, where is the projection angle of an x-ray which produces an attenuation value as measured from the midpoint of the fan-beam.
 4. A method for estimating fan-beam projection data acquired from a subject at a selected projection angle, the steps comprising: a) acquiring fan-beam projection data from the subject at other projection angles; and b) estimating the fan-beam projection data from the acquired fan-beam projection data using a fan-beam data consistency condition.
 5. The method as recited in claim 4 in which the fan-beam projection data is an x-ray attenuation profile acquired with a CT imaging system.
 6. The method as recited in claim 4 in which step b) includes: i) filtering the acquired fan-beam projection data with a first filtering kernel; ii) rebinning the filtered fan-beam projection data; and iii) calculating estimates of the fan-beam projection data by filtering rebinned projection data with a second filtering kernel.
 7. The method as recited in claim 6 in which the second filtering kernel is a Hilbert kernel.
 8. The method as recited in claim 6 in which the first filtering kernel is I/sinγ, where γ is the projection angle of a beam which produces a value in the projection data as measured from the midpoint of the fan-beam. 